We consider two-level preconditioning schemes in which the coarse-level spaces are defined on the same geometrical mesh as the fine-level discontinuous space. Namely, we study coarse spaces based on continuous piecewise polynomials and based on piecewise constants. We show that these two techniques give rise to uniform preconditioners and present numerical evidence confirming the theoretical results. We also consider multilevel extensions of the two-level methods for mesh hierarchies obtained by either uniform mesh refinement of a coarse-grid mesh or element agglomeration of an ustructured fine-grid mesh. We present some theoretical results for these methods as well as numerical experiments based on them.
This work was performed in collaboration with R. Lazarov, P. Vassilevski, and L. Zikatanov.